US8024387B2 - Method for synthesizing linear finite state machines - Google Patents
Method for synthesizing linear finite state machines Download PDFInfo
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- US8024387B2 US8024387B2 US11/894,393 US89439307A US8024387B2 US 8024387 B2 US8024387 B2 US 8024387B2 US 89439307 A US89439307 A US 89439307A US 8024387 B2 US8024387 B2 US 8024387B2
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
- G06F7/58—Random or pseudo-random number generators
- G06F7/582—Pseudo-random number generators
- G06F7/584—Pseudo-random number generators using finite field arithmetic, e.g. using a linear feedback shift register
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03K—PULSE TECHNIQUE
- H03K3/00—Circuits for generating electric pulses; Monostable, bistable or multistable circuits
- H03K3/84—Generating pulses having a predetermined statistical distribution of a parameter, e.g. random pulse generators
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2207/00—Indexing scheme relating to methods or arrangements for processing data by operating upon the order or content of the data handled
- G06F2207/58—Indexing scheme relating to groups G06F7/58 - G06F7/588
- G06F2207/583—Serial finite field implementation, i.e. serial implementation of finite field arithmetic, generating one new bit or trit per step, e.g. using an LFSR or several independent LFSRs; also includes PRNGs with parallel operation between LFSR and outputs
Definitions
- Linear finite state machines such as linear feedback shift registers (LFSRs) and cellular automata (CA) are often used for generating pseudo-random sequences.
- LFSMs linear feedback shift registers
- CA cellular automata
- An LFSR includes memory elements such as flip-flops and linear logic gates such as XOR or XNOR gates connected as shown in FIG. 1 .
- the combined (added modulo 2) output of each stage is fed back to the first stage of the LFSR.
- Such an implementation is called a type I LFSR or Fibonacci generator.
- a nonzero n-bit vector (frequently called a seed) is loaded into the register, and a clock is pulsed at the appropriate rate.
- An LFSR initialized as described above can cycle through a number of states before coming back to the initial state. If an n-bit LFSR can cycle through all possible 2 n ⁇ 1 nonzero states, then its characteristic polynomial is called a primitive characteristic polynomial.
- Such an LFSR is often referred to as a maximum-length LFSR, and the resultant output sequence is termed a maximum-length sequence or m-sequence.
- M-sequences have a number of unique properties, as described in P. H. Bardell, W. H. McAnney, and J. Savir, Built - In Test for VLSI: Pseudorandom Techniques , John Wiley & Sons, 1987.
- FIG. 2 An alternative LFSR implementation is shown in FIG. 2 . It is called a type II LFSR or Galois true divisor. A distinct feature of this implementation is that the output of the last stage of the LFSR is fed back to prior stages as indicated by the characteristic polynomial employed. As with a type I LFSR, a type II LFSR constructed in accordance with a primitive characteristic polynomial and loaded with a nonzero n-bit vector will produce all 2 N ⁇ 1 nonzero states.
- LFSMs such as the LFSRs described above are employed in a vast variety of applications, including error detection and correction, data transmission, mobile telephony, cryptography, testing of very large scale integrated circuits, data compression, and hardware white noise generation. For high-performance applications, the required data generation and compression can only be achieved by high-performance circuits.
- U.S. Pat. No. 5,268,949 describes a pseudo-random test pattern generator having a higher operating speed than the conventional LFSRs.
- the speed of any LFSR is determined by the performance of the respective elements comprising the generator.
- an XOR feedback network may introduce significant delays if an LFSR features a large number of feedback taps.
- the patent proposes the use of a number of LFSRs connected in parallel fashion and operated at lower clock speed, wherein the actual output signals are produced on the multiplex basis.
- this architecture has much larger area overhead than typical LFSRs and its performance is still limited by multiplexers in the output function.
- U.S. Pat. No. 5,412,665 describes another parallel-operation high-speed test pattern generation and test response compaction implemented by means of low-speed components. It utilizes a number of flip-flops and connected combinational logic networks. Each network provides a pseudo-random pattern which is output in parallel, thereby creating a high-speed data flow with an increased area of the circuitry.
- U.S. Pat. No. 5,466,683 describes a programmable LFSR that produces pseudo-random patterns having a variable characteristic polynomial. Its sequence generator is driven by applying appropriate control signals in such a way that a given control signal has a value of 1 if the corresponding term of the characteristic polynomial has a coefficient of 1. Consequently, the proposed scheme places an XOR gate and associated two-way multiplexer in the front of each LFSR stage, thus incurring significant area overhead.
- a similar architecture of a programmable LFSR with a provision for an initialization circuitry is given in U.S. Pat. No. 5,090,035.
- U.S. Pat. No. 5,790,626 describes a bi-directional LFSR employing latches having dual (forward and reverse) inputs.
- This LFSR can generate both state trajectories: the forward sequence of patterns, and the reverse one corresponding to an original feedback polynomial and its reciprocal counterpart, respectively.
- the register features two different linear feedback networks that operate exclusively at a time, but can intermix forward and reverse steps, thus allowing the vector generation process to follow the native sequence of the LFSR in both directions at any time.
- a similar concept is also disclosed in the U.S. Pat. No. 5,719,913, where the XOR gates in the feedback network are driven by multiplicity of two-way multiplexers.
- a method for synthesizing a linear feedback shift register includes the following steps.
- An original linear finite state machine circuit is obtained, the circuit including a plurality of memory elements and linear logic gates and capable of generating an output sequence.
- Feedback connections in the original circuit are determined, a feedback connection spanning a number of memory elements and including a source tap and destination tap connected by an associated feedback connection line.
- the source and destination taps of one or more of the feedback connection are then shifted across a number of memory elements in the same direction.
- a method for synthesizing a linear finite state machine includes the following steps.
- An original linear finite state machine circuit is obtained, the circuit including a plurality of memory elements and linear logic gates and capable of generating an output sequence.
- At least first and second feedback connections in the original circuit are determined, each feedback connection spanning a number of memory elements and including a source tap and destination tap connected by an associated feedback connection line, the destination tap including a destination linear logic gate.
- the source and destination taps of the feedback connections are then shifted relative to one another such that the destination tap of the first feedback and the source tap of the second feedback connection cross.
- Another feedback connection line is then added between a source tap of the first feedback connection and a destination linear logic gate at a destination tap of the second feedback connection.
- a linear finite state machine circuit comprises a plurality of memory elements and linear logic gates, wherein fan-out within the circuit is no greater than two and the number of level of linear logic within the circuit is no greater than one.
- FIG. 1 is a diagram of a type I LFSR.
- FIG. 2 is a diagram of a type II LFSR.
- FIG. 3 is a flowchart of a first synthesis method in accordance with the invention.
- FIGS. 4A and B illustrate an EL transformation of an LFSR in accordance with the method.
- FIG. 5 is a flowchart of a method for initializing LFSRs in accordance with the invention.
- FIGS. 6A and B illustrate application of an elementary shift to the left (EL) transformation that causes two linear logic gates in an LFSR to cross.
- EL elementary shift to the left
- FIGS. 7A and B illustrate application of an E elementary shift to the right (E) transformation that causes two source taps in an LFSR to cross.
- FIG. 8 is a flowchart of a second synthesis method in accordance with the invention.
- FIGS. 9A-C illustrate application of a source tap crossing a destination tap while moving to the left (SDL) transformation in accordance with the second method.
- FIGS. 10A-C illustrate application of a source tap crossing a destination tap while moving to the right (SDR) transformation in accordance with the second method.
- FIGS. 11A-C illustrate application of a destination tap crossing a source tap while moving to the left (DSL) transformation in accordance with the second method.
- FIGS. 12A-C illustrate application of a destination lap crossing a source tap while moving to the right (DSR) transformation in accordance with the second method.
- FIGS. 13A-D are examples of synthesizing an LFSR from three types of LFSMs: a type I LFSR, a type II LFSR, and a linear cellular automaton.
- FIGS. 14A-D are an example of synthesizing an LFSR by applying a combination of different transformations to an original LFSR circuit.
- methods for synthesizing LFSMs in accordance with the invention are implemented in software stored on a computer-readable medium and executed on a general-purpose computer system.
- a computer system is represented by block 18 in FIG. 3 .
- the invention for example, can be implemented in computer aided-design tools that explore the domain of possible solutions and different trade-offs concerning the layout of LFSRs. For clarity, only those aspects of the software germane to the invention are described; product details well known in the art are omitted. For the same reason, the computer hardware is not described in further detail. It should appreciated that the invention is not limited to use with computer system 18 or any particular computer language or program.
- FIG. 2 shows an LFSM in the form of an arbitrary maximum-length type II LFSR with n memory elements such as flip-flops or latches and a number of feedback connections.
- Each feedback connection includes a source tap corresponding to an output of a memory element feeding this particular connection, a feedback connection line spanning a number of memory elements as defined by the primitive characteristic polynomial employed, and a linear gate such as an XOR gate placed at a destination tap of the feedback connection, that is, at the input to another memory element.
- the LFSR architecture can be transformed by shifting its feedback connections across memory elements for the purpose of performance optimization and to minimize the total length of the feedback lines.
- transformations may be carried out in such a way that they preserve the m-sequence of the original LFSR circuit, although the modified LFSR circuit may feature a different state trajectory than that of the original circuit. That is, the LFSR state trajectories (the contents of the memory elements at any given time) may differ between the original and modified circuits although the m-sequence, taken from an output of each circuit, is preserved. If the same LFSR seed is used in both circuits, then the m-sequence is the same when taken from different memory elements. If different LFSR seeds are available, then the m-sequence may be the same when taken from the same memory element.
- FIG. 3 is a flowchart of a first synthesis method
- FIGS. 4A and B illustrate an application of the method to an LFSR transformation called an elementary shift to the left, or EL.
- FIG. 4A shows the original LFSR circuit with a feedback connection 20 spanning a number of memory elements and including a source tap 22 at the output of memory element Z and a destination tap (including a destination XOR gate 24 ) at the input to memory element C. The taps are connected by an associated feedback connection line.
- FIG. 4B shows the modified LFSR circuit resulting from the transformation.
- all memory elements but the rightmost one (Z) are assumed to contain initially symbols a, b, c, . . . , p.
- the memory element Z should initialized to 0 (or initialized to 1 if an XNOR gate is used in place of the XOR gate).
- the memory elements After one shift ( FIG. 4A ), the memory elements contain symbols d, a, b, . . . , q, p, as a new symbol d enters the memory element A.
- the contents of the memory elements are as follows: e, d, a ⁇ p, . . . , r, q. Further operation of the LFSR produces additional shifts of data as shown. Now, in FIG.
- a transformation EL is applied to the original LFSR circuit, and it places the XOR gate 24 at the input of the memory element B and relocates the source tap 22 of the feedback connection 20 to the output of memory element Y, accordingly.
- i can be observed that the contents of the memory elements spanned by the original feedback line, that is, flip-flops C, . . . , Y, Z, match the symbols appearing at the outputs of flip-flops C, . . . , Y, Z in the original circuit.
- a copy of the original LFSR circuit is obtained by synthesis software or an equivalent tool ( 26 ), typically from secondary storage or from memory if entered directly by a user.
- the feedback connections in the original circuit are then determined ( 28 ), such as the feedback connection spanning memory elements C through Z in the LFSR circuit of FIG. 4A .
- One or more of the feedback connection may then shifted across a number of memory elements in the original circuit in the same direction ( 30 ). These shifts are carried out to reduce the length of feedback lines, to reduce the levels of linear logic, and to reduce the internal fan-out of the original circuit.
- FIG. 5 is a flowchart that illustrates how, by selecting an appropriate seed, the m-sequence can be preserved in the modified LFSR circuit despite the shifting of feedback connections across memory elements.
- the direction of shift is determined ( 32 )—left (defined as upstream, against the direction of data flow through the memory elements) or right (defined as downstream, with the direction of data flow through the memory elements).
- the initial LFSR vector, or seed is provided with the same logic values for memory elements being shifted out of the feedback, connection as a result of the shift.
- the initial LFSR vector, or seed is provided with the same logic values for memory elements being shifted into the feedback connection as a result of the shift.
- the same logic values are zero if the linear gates of the original circuit are XOR gates and the same logic values are one if the linear gates of the original circuit are XNOR gates.
- Transformations EL and ER can be extended to handle cases in which a destination gate (or a source tap) of a feedback connection being moved crosses another destination gate (or source tap), respectively. Examples of these situations are illustrated in FIGS. 6A and B and 7 A and B.
- the internal (shorter) feedback connections 40 and 42 in FIGS. 6A and 7A can be shifted to the left or to the right in FIGS. 6B and 7B , respectively, and no further transformations are required.
- the shifted feedback connection provides symbols to memory elements whose contents remain unaffected by transformations EL or ER.
- This form of the transformations thus preserves the maximum-length property of the circuit, provided that all memory elements are initialized with an appropriate seed as described above.
- flip-flop Q in FIGS. 6A and B and flip-flop Y in FIGS. 7A and B should be initialized to 0 when performing transformations EL and ER, respectively.
- FIG. 8 is a flowchart that illustrates a second synthesis method wherein a feedback connection shift causes the destination gate in one feedback connection and the source tap in another feedback connection to cross.
- the method can be used if the original LFSR circuit has at least two feedback connections ( 50 ).
- the circuit topology is checked after a shift to determine if a destination gate and a source tap have crossed ( 52 ). If not, the first method continues to its conclusion ( 54 ). However, if a destination gate and source tap cross, an appropriate feedback connection is added to the LFSR circuit ( 56 ) as described below.
- SDL a source tap crosses a destination gate while moving to the left
- SDR a source tap crosses a destination gate while moving to the right
- DSL a destination gate crosses a source tap while moving to the left
- DSR a destination gate crosses a source tap while moving to the right
- Transformation SDL is illustrated in FIGS. 9A-C . It can be used when two feedback connections 58 and 59 are arranged in such a way that a linear gate 60 (such as the XOR gate shown) at the destination tap of the first feedback connection is separated from a source tap 62 of the second feedback connection by a single memory element, as shown in FIG. 9A . During the first part of the transformation, the source tap 62 shifts across this memory element ( FIG. 9B ). The XOR gate 64 at the destination tap of the second feedback connection also shifts to the left accordingly. This operation preserves the maximum-length property of the LFSR since this act is equivalent to transformation EL described earlier.
- a linear gate 60 such as the XOR gate shown
- symbol a must be provided by the source tap 66 of the first feedback connection 58 to the XOR gate 64 . This is accomplished by adding a feedback connection line 68 between the source tap 66 and the XOR gate 64 at the shifted destination tap. It is worth noting that symbol a can represent several feedback paths reaching their destination at this particular gate. In such a case, all of these feedback connections should be extended as required by transformation SDL. The same rule applies to transformations SDR, DSL, and DSR.
- Transformation SDR is shown in FIGS. 10A-C .
- both feedback connections 78 and 79 involved in this operation do not span any common memory elements ( FIG. 10A ).
- the second feedback connection 79 to be shifted to the right, has its source tap 82 at the output of the flip-flop feeding the XOR gate 80 at the destination tap of the first feedback connection 78 . Therefore, the output of the gate 80 is equal to a ⁇ b.
- the source tap 82 crosses the XOR gate 80 , thus changing functionality of the circuit ( FIG. 10B ).
- a feedback connection line 88 is added between the XOR gate 84 and the source tap 86 of the first feedback connection 78 .
- an ER transformation may be carried out on the second feedback connection 79 with no effect on the function of the LFSR, the transformation adding an additional XOR gate 89 ( FIG. 10C ).
- Transformation DSL is shown in FIGS. 11A-C .
- the initial setup ( FIG. 11A ) as well as the first acts are similar to those of transformation SDR. Consequently, a new feedback connection line 90 is added to restore an original functionality of the circuit ( FIG. 11B ).
- a transformation EL is performed on the first feedback connection 92 , leading to a structure with XOR gate 94 of the first feedback connection shifted by one memory element to the left.
- Transformation DSR is shown in FIGS. 12A-C .
- transformation ER is first applied to the first feedbck connection 100 ( FIG. 12B ).
- the XOR gate 102 of the first feedback connection is shifted such that it crosses the source tap 104 of the second feedback connection 106 , or equivalently, the source tap 104 is moved from the output of the XOR gate 102 to the gate's input ( FIG. 12 c ).
- This last act removes symbol b from the sum a ⁇ b being provided to the XOR gate 108 of the second feedback connection. Its loss must be compensated for by adding a feedback connection line 110 between the source tap 112 of the first feedback connection 100 and the XOR gate 108 to maintain both arguments, a and b, on the gate's inputs ( FIG. 9C ).
- the transformations described can be utilized one or more times in synthesizing a LFSM. They can also be combined with other transformations in a synthesis. Examples of these possible applications are described below.
- the architecture of the modified linear finite state machine that can be obtained from these transformations is characterized by an internal fan-out no greater than two, no more than one level of linear logic gates, and short feedback connection lines.
- FIGS. 13A-D are examples of synthesizing an LFSR from various types of LFSMs, including a type I LFSR, a type II LFSR, and a linear cellular automaton, by successive applications of EL transformations.
- the structure of the LFSR shown in FIG. 13A is a true Galois divisor or type II shift register implementing primitive characteristic polynomial x 32 +x 30 +x 21 +x 16 +x 11 +x 4 +1, with five feedback connections that includes lines 120 - 128 each connecting a shared source tap 129 to separate destination taps that include XOR gates 130 - 138 , respectively.
- the XOR gates are each disposed in a respective forward transmission path along the chain of memory elements.
- each XOR gate has one input coupled to the output of a preceding stage, its output coupled to the input of the succeeding stage, and a second input connected to the feedback path line originating at the output of memory element 0 .
- the overall layout of the LFSR circuit has been optimized prior to any further transformations by forming a ring structure. Nevertheless, two of the most significant benefits of the present synthesis methods appear in FIG. 13D , which illustrates a transformation of the original type II LFSR circuit of FIG. 13A to a modified LFSR circuit. As can be seen, the modified LFSR of FIG.
- 13D has been obtained by applying the transformation EL to the five feedback connections (represented by coefficients x 30 , x 21 , x 16 , x 11 , and x 4 ) one, five, eight, ten, and fourteen times, respectively.
- the internal fan-out of the LFSR has also been reduced by a factor of three, from six elements (memory element 31 and the five XOR gates 130 - 138 ) fed by flip-flop 0 in the original LFSR circuit to only two elements (the next memory element and one XOR gate) fed by any flip-flop in the modified LFSR circuit. Furthermore, the modified LFSR circuit of FIG. 13D has, in its worst case, only one level of XOR logic between any pair of flip-flops.
- An LFSR can also be synthesized from other types of LFSMs.
- the modified LFSR shown of FIG. 13D can be obtained from the type I LFSR of FIG. 13A (implementing the same primitive characteristic polynomial x 32 +x 30 +x 21 +x 16 +x 11 +x 4 +1) by applying the transformations described above.
- the modified LFSR of FIG. 13D can be obtained from the 32-bit linear cellular automaton of FIG. 13C (implementing also the same primitive characteristic polynomial x 32 +x 30 +x 21 +x 16 +x 11 +x 4 +1) by applying these transformations with null boundary conditions shown in the figure.
- FIGS. 14A-D are an example of synthesizing an LFSR by applications of a combination of the above transformations, in this case EL transformations and an SDL transformation.
- FIG. 14A depicts a type II LFSR implementing primitive characteristic polynomial x 8 +x 6 +x 5 +x+1.
- Applying the transformation EL four times to the feedback connection represented by coefficient x leads to the circuit shown in FIG. 14B .
- Applying transformation SDL then shifts feedback connection 130 further to the left by one memory element and adds a feedback connection line 136 at the input to the XOR gate 134 ( FIG. 14C ).
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US13/013,712 US8533547B2 (en) | 1999-11-23 | 2011-01-25 | Continuous application and decompression of test patterns and selective compaction of test responses |
US14/021,800 US9134370B2 (en) | 1999-11-23 | 2013-09-09 | Continuous application and decompression of test patterns and selective compaction of test responses |
US14/853,412 US9664739B2 (en) | 1999-11-23 | 2015-09-14 | Continuous application and decompression of test patterns and selective compaction of test responses |
US15/608,716 US10234506B2 (en) | 1999-11-23 | 2017-05-30 | Continuous application and decompression of test patterns and selective compaction of test responses |
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US09/957,701 US6539409B2 (en) | 1999-11-23 | 2001-09-18 | Method for synthesizing linear finite state machines |
US10/346,699 US6708192B2 (en) | 1999-11-23 | 2003-01-16 | Method for synthesizing linear finite state machines |
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